Learn: a square cut along its diagonal
A square tile has side length 4 cm. Cutting it along a diagonal makes two equal right triangles. Each triangle has two 4 cm legs. How long is the diagonal?
Name the sides before choosing a ratio
A right angle measures 90°. The two sides meeting there are the legs. The side opposite that angle is the hypotenuse, the longest side. The square-root symbol √ means the nonnegative number whose square equals the number inside it; √2 × √2 = 2.
Why a 45–45–90 triangle has ratio 1 : 1 : √2
The two acute angles are equal, so the legs are equal. Let each leg have length s, with s > 0. The Pythagorean theorem gives c² = s² + s² = 2s². Taking the positive square root gives c = s√2. Thus legs s and s have hypotenuse s√2.
For the tile, c = 4√2 cm, approximately 5.66 cm. The exact answer keeps √2; the decimal is rounded. Do not use 4 + 4 = 8 cm for the diagonal.
Why a 30–60–90 triangle has ratio 1 : √3 : 2
Split an equilateral triangle of side 2s into two equal right triangles by drawing an altitude to the midpoint of its base. Each half has a short leg s opposite 30°, hypotenuse 2s, and long leg opposite 60°. By the Pythagorean theorem, the long leg squared is (2s)² − s² = 3s², so its length is s√3.
The order matters: opposite 30° → s; opposite 60° → s√3; opposite 90° → 2s. The short leg is half the hypotenuse, not half the long leg.
Explore: match each side to its angle
The diagrams below have the correct side proportions. Their labels show exact lengths for s = 1; they are reference drawings, not measurements to take from your screen.
Sides are in ratio units; no physical unit is assigned to these reference drawings.
s = 4 cm: legs 4 cm and 4 cm; hypotenuse 4√2 cm (about 5.66 cm).
Worked example: start from the long leg
A 30–60–90 triangle has long leg 6√3 cm. Match it to s√3, so s√3 = 6√3 and s = 6. The short leg is 6 cm and the hypotenuse is 12 cm. Check: 6² + (6√3)² = 36 + 108 = 144 = 12².
Practice: identify the given side
1. A 45–45–90 triangle has a 7 cm leg. Find the other sides.
The other leg is 7 cm; the hypotenuse is 7√2 cm.
2. A 30–60–90 triangle has hypotenuse 10 m. Find its legs.
2s = 10, so s = 5. Short leg 5 m; long leg 5√3 m.
3. A 45–45–90 triangle has hypotenuse 8√2 cm. Find each leg.
s√2 = 8√2, so each leg is 8 cm.
4. Can these ratios be used for any right triangle?
No. Confirm the acute angles or equivalent side relationships first. A general right triangle still satisfies a² + b² = c², but need not have either special ratio.
Review: angle, side, scale
Mark the right angle, identify the hypotenuse, match the given side to the correct part of the ratio, then find s. Retain units and distinguish exact answers from rounded ones. Explore geometry or precalculus.
Optional reference: OpenStax: right-triangle relationships.